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Properties

Label 2-465-465.284-c0-0-0
Degree $2$
Conductor $465$
Sign $0.920 + 0.390i$
Analytic cond. $0.232065$
Root an. cond. $0.481731$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + (−0.5 + 0.866i)3-s + (−0.5 − 0.866i)5-s + (0.5 − 0.866i)6-s + 8-s + (−0.499 − 0.866i)9-s + (0.5 + 0.866i)10-s + 0.999·15-s − 16-s + (0.5 − 0.866i)17-s + (0.499 + 0.866i)18-s + (0.5 − 0.866i)19-s + 2·23-s + (−0.5 + 0.866i)24-s + (−0.499 + 0.866i)25-s + ⋯
L(s)  = 1  − 2-s + (−0.5 + 0.866i)3-s + (−0.5 − 0.866i)5-s + (0.5 − 0.866i)6-s + 8-s + (−0.499 − 0.866i)9-s + (0.5 + 0.866i)10-s + 0.999·15-s − 16-s + (0.5 − 0.866i)17-s + (0.499 + 0.866i)18-s + (0.5 − 0.866i)19-s + 2·23-s + (−0.5 + 0.866i)24-s + (−0.499 + 0.866i)25-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 465 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.920 + 0.390i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 465 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.920 + 0.390i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(465\)    =    \(3 \cdot 5 \cdot 31\)
Sign: $0.920 + 0.390i$
Analytic conductor: \(0.232065\)
Root analytic conductor: \(0.481731\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{465} (284, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 465,\ (\ :0),\ 0.920 + 0.390i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3715523007\)
\(L(\frac12)\) \(\approx\) \(0.3715523007\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.5 - 0.866i)T \)
5 \( 1 + (0.5 + 0.866i)T \)
31 \( 1 - T \)
good2 \( 1 + T + T^{2} \)
7 \( 1 + (0.5 + 0.866i)T^{2} \)
11 \( 1 + (0.5 - 0.866i)T^{2} \)
13 \( 1 + (0.5 - 0.866i)T^{2} \)
17 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
19 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
23 \( 1 - 2T + T^{2} \)
29 \( 1 - T^{2} \)
37 \( 1 + (0.5 + 0.866i)T^{2} \)
41 \( 1 + (0.5 - 0.866i)T^{2} \)
43 \( 1 + (0.5 + 0.866i)T^{2} \)
47 \( 1 + T + T^{2} \)
53 \( 1 + (1 + 1.73i)T + (-0.5 + 0.866i)T^{2} \)
59 \( 1 + (0.5 + 0.866i)T^{2} \)
61 \( 1 + T + T^{2} \)
67 \( 1 + (0.5 - 0.866i)T^{2} \)
71 \( 1 + (0.5 - 0.866i)T^{2} \)
73 \( 1 + (0.5 - 0.866i)T^{2} \)
79 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
83 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
89 \( 1 - T^{2} \)
97 \( 1 - T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−11.15656452755981616471003055364, −10.06405642542043154611307794102, −9.343060626975286453602776696083, −8.820318729542164717026879776912, −7.84649008211341395229316268960, −6.77827118660941749134615733295, −5.04180868395645011305950218030, −4.81257799002843212983593466377, −3.33673869767187524700536486229, −0.856109940499512928772712699344, 1.37689400009198586643998599742, 3.07974202564348208892737845561, 4.66394389673503000949777505746, 6.01329707078007513160472847820, 7.00010126507466266496157769492, 7.74981581196623139682372121176, 8.360369296561677572553263722885, 9.559971077703776778472959618559, 10.58668686682659566959015912902, 11.02619647633740087109771981914

Graph of the $Z$-function along the critical line